# Do retrograde spin-orbit resonances exist?

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The end state of rotation of an initially fast-spinning prograde terrestrial planet (in the absence of additional forcings such as "thermal tides" in an atmosphere, e.g. Venus) is a spin-orbit resonance of the form $$n + 1 : 2, n in {1,2,3,… }$$. Examples being the common 1:1 resonance ($$n=1$$), and the 3:2 resonance of Mercury ($$n=2$$). See also this paper (Correia & Laskar, 2001) discussing the probability of Mercury's capture into higher-order resonances, e.g. 2:1 or 5:2.

What about the case for a planet with retrograde rotation? Do there exist retrograde spin-orbit resonances, e.g. extending the sequence to negative $$n$$, giving resonances such as $$-1:1$$ or $$-3:2$$, and would there be any probability of capture into such a state?

## Do retrograde spin-orbit resonances exist? - Astronomy

OK, this is something of a ‘marker posting’. Not a lot of ‘great insight’ but marking the existence of something I’ve been trying to show “exists” for a couple of years now. That is “Spin Orbit Coupling” at the macro level.

Substantially every time I’ve got on a Google Train of “Spin Orbit Coupling” it’s pulled into a sub atomic station. It’s just all over the place at that scale. You can’t hardly say “particle” without some kind of spin orbit coupling effect popping up.

Physics (like the conservation of angular momentum) implied to me that this is a property that ought to exist at all physical scales. I could not imagine a way to have angular momentum act differently at the sub atomic scale than at the macro… Yet despite my expectation, I could find almost nothing that said “spin orbit coupling” in the context of “planet”. The one exception being a paper by Ian Wilson that showed a Length Of Day (LOD) correlation with the PDO and correlation with the solar system orbital state. Yet “mainstream” climate folks were not keen on that paper, so I wanted ‘more than one’ reference. That, and the lack of other references was a bit unnerving.

But now I’ve run into a couple of more examples.

### Why Does This Matter?

So, who cares? Well, the whole complaint about the “Solar output modulated by planets” thesis comes down to “only tides can happen and they are too small to matter”. But if there can be spin-orbit couples, then the entire Angular Momentum of the outer planets can have impact. As Angular momentum increases greatly with Radius, this means they could have a lot of ‘pull’ -) (Please, it’s only a little pun…)

Basically, if Spin Orbit Coupling is active and recognized at the macro / planet level, then folks need to address the question of what is happening to solar SPIN when the Angular Momentum of the solar system changes as the barycenter moves into and out of the sun. Having a spin change (even, or perhaps especially) if concentrated into particular layers or bands, could easily explain planetary modulation of solar output.

### So where did I find a reference?

First, the bad news: It’s largely Wikipedia.

Second, the good news: The articles are not contentious, so ought to be fairly politically neutral. (Though now that I’ve tied them to Global Warming there is the risk that the Political Thought Police of AGW will go and erase or re-write them.)

For an example of the typical, highly subatomic particle oriented article, there is this one:

Just full of stuff that sounds very sub-atomic specific and particle centric:

In quantum physics, the spin-orbit interaction (also called spin-orbit effect or spin-orbit coupling) is any interaction of a particle’s spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron’s atomic energy levels due to electromagnetic interaction between the electron’s spin and the nucleus’s magnetic field. This is detectable as a splitting of spectral lines. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin-orbit effects for electrons in semiconductors and other materials are explored and put to useful work.

Then here are two places where I’ve found a non-particle level reference:

While much of that article is about sub-atomic spin-orbit coupling, it contained this giant gem:

In astronomy, spin-orbit coupling reflects the general law of conservation of angular momentum, which holds for celestial systems as well.

EXACTLY what I was looking for. My understanding that physics of momentum is the same at any and ALL scales, reflected in this one statement. I know, I ought not to have been so tentative about it and I ought to have just baldly asserted that it was the case. But there are enough things that are ‘different’ in the sub-atomic and quantum worlds that I was leery of just leaping from that to the Galactic without a bit of moral support / confirmation. It goes on:

In simple cases, the direction of the angular momentum vector is neglected, and the spin-orbit coupling is the ratio between the frequency with which a planet or other celestial body spins about its own axis to that with which it orbits another body. This is more commonly known as orbital resonance. Often, the underlying physical effects are tidal forces.

This lead to the insight that there was a ‘name change’ hiding the connection. That in astronomy, the concept was being hidden in / blended with “Orbital Resonance”. So here we have The Giant Missing Clue. The name was changed. But the point is clearly made that ‘body spin” can be interchanged with orbital rotation. Just like the earth spin is being changed by the moon via tides. I note in passing that it says “Often” … “effects are tidal forces.” (but not always?)

Also of note is that statement that the angular momentum vector is neglected in “simple cases”. Astronomy seems over full of “neglecting” and “simple cases”. In particular I note that they say when the momentum vector is ignored it is often called “orbital resonance”. OK, but what if I don’t want to ignore that vector? This, I think, is where the improvement will come. Just apply straight Angular Momentum physics, but do all of it this time…

But at least now we have something (even if small) to point at when asserting that maybe, just maybe, that whole Solar System Barycenter Conservation Of Planetary Angular Momentum thing could actually stir the sun up a bit. Be it “spin”, or “tides”, or “nutation”, or “precession”, or …

The problem is no longer “Can it?”. The problem is now “What happens to the COMBINED angular momentum of ALL the planets AND the sun as orbital positions change? How does this change the solar motions?”. And given that Angular Momentum is dominated by the Radius, there is A LOT of angular momentum to spread around.

Types of resonance

In general, an orbital resonance may:

involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus orbit inclination).

act on any time scale from short term, commensurable with the orbit periods, to secular, measured in 104 to 106 years.

lead to either long term stabilization of the orbits or be the cause of their destabilization.

So any orbital feature can be involved, and that includes things like pole precession, nutation (“wobble”), etc. And this implied, through them, to spin.

A Lindblad resonance drives spiral density waves both in galaxies (where stars are subject to forcing by the spiral arms themselves) and in Saturn’s rings (where ring particles are subject to forcing by Saturn’s moons).

A secular resonance occurs when the precession of two orbits is synchronised (usually a precession of the perihelion or ascending node). A small body in secular resonance with a much larger one (e.g. a planet) will preecess at the same rate as the large body. Over long times (a million years, or so) a secular resonance will change the eccentricity and inclination of the small body.

I note in passing that “forcing” is starting to show up here, as well. IFF they mean “force” they ought to say so. If they mean a mathematical “forcing function” they ought to say that instead (and state the “given function” so we know what function they are talking about…)

Several prominent examples of secular resonance involve Saturn. A resonance between the precession of Saturn’s rotational axis and that of Neptune’s orbital axis (both of which have periods of about 1.87 million years) has been identified as the likely source of Saturn’s large axial tilt (26.7°). Initially, Saturn probably had a tilt closer to that of Jupiter (3.1°). The gradual depletion of the Kuiper belt would have decreased the precession rate of Neptune’s orbit eventually, the frequencies matched, and Saturn’s axial precession was captured into the spin-orbit resonance, leading to an increase in Saturn’s obliquity. (The angular momentum of Neptune’s orbit is 104 times that of that of Saturn’s spin, and thus dominates the interaction.)

So Saturn’s ’tilt’ is being driven by the coupling of its spin to Neptune’s orbital precession.

I’m sure there are many more to be found, now that the jargon mismatch has been found out. Now that we see that in astronomy it’s called ‘resonance’ while everyone else calls it ‘spin orbit coupling’.

### Some Notes On Angular Momentum

The wiki on Angular Momentum makes it clear why this all matters.

They also have a less math heavy and less technical intro to angular momentum here:

In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system. The angular momentum L of a particle with respect to some point of origin is

L = r x p
or
L = r x mv

where r is the particle’s position from the origin, p = mv is its linear momentum, and × denotes the cross product.

The key bits are that it’s a conserved property. That means angular momentum doesn’t just go away. You’ve got to turn it into something else. That it is directly proportional to radius (distance from the origin) means that smaller things can have a lot of angular momentum if they are a long ways away. Yes, Mass (m) matters, but make that Radius (r) longer and the mass can be smaller with the same impact. So the sun has a whole lot of Mass, but it’s Radius of orbit about the barycenter is very small. Saturn is much smaller, but oh does it have a long lever arm to work through.

The angular momentum of a system of particles (e.g. a rigid body) is the sum of angular momenta of the individual particles. For a rigid body rotating around an axis of symmetry (e.g. the fins of a ceiling fan), the angular momentum can be expressed as the product of the body’s moment of inertia I (a measure of an object’s resistance to changes in its rotation rate) and its angular velocity ω:

In this way, angular momentum is sometimes described as the rotational analog of linear momentum.

Angular momentum is conserved in a system where there is no net external torque, and its conservation helps explain many diverse phenomena. For example, the increase in rotational speed of a spinning figure skater as the skater’s arms are contracted is a consequence of conservation of angular momentum. The very high rotational rates of neutron stars can also be explained in terms of angular momentum conservation. Moreover, angular momentum conservation has numerous applications in physics and engineering (e.g. the gyrocompass).

There is just no getting away from the need to conserve angular momentum. Period. Also notice the emphasis on ‘rigid body’. But the sun is not rigid. I suspect that is the great mistaken simplification.

So when the solar masses change their positions relative to the barycenter (center of rotation of the combined solar system and the center of the sun’s “orbit”) the sun suffers a change of Angular Momentum as it’s Radius (r) has changed. And that must show up somewhere.

The question now becomes “Where does it go?”…

As the coupling can, per all I can find, happen at any scale and distance, there are many places it can go. But there is one place it can not go, and that is ‘away’. I suppose it could even end up in subatomic spin (though I can’t imagine how) or out in the outer planets as orbital perturbations. My ‘best guess’ would be that there is some minor change of the flow of “currents” on the Sun. Perhaps the Solar Conveyor Belt slowing down is the consequence? Or some other modulation such that the Landscheidt predictions have a direct mechanism.

The simple fact is that the sun is about 1 % of the solar system angular momentum. The changes “out there” are far larger and more important than its whole package. And everyone is simply ignoring that.

Though I do note that this page on the formation of the solar system talks about angular momentum exchanges with some frequency:

Among the extrasolar planets discovered to date are planets the size of Jupiter or larger but possessing very short orbital periods of only a few days. Such planets would have to orbit very closely to their stars so closely that their atmospheres would be gradually stripped away by solar radiation. There is no consensus on how to explain these so-called hot Jupiters, but one leading idea is that of planetary migration, similar to the process which is thought to have moved Uranus and Neptune to their current, distant orbit. Possible processes that cause the migration include orbital friction while the protoplanetary disc is still full of hydrogen and helium gas and exchange of angular momentum between giant planets and the particles in the protoplanetary disc.

For what it’s worth, looking at a google of “Solar orbital resonance” is more fruitful than the “spin orbit coupling” term, though it leads to a large number of articles describing the familiar cases such as orbital locking (The Moon always presenting the same face to earth).

has an abstract for a paper from Cornell University Library:

Orbital Resonance and Solar Cycles
P.A.Semi
(Submitted on 29 Mar 2009)
We present an analysis of planetary moves, encoded in DE406 ephemerides.
We show resonance cycles between most planets in Solar System, of differing quality. The most precise resonance – between Earth and Venus, which not only stabilizes orbits of both planets, locks planet Venus rotation in tidal locking, but also affects the Sun:
This resonance group (E+V) also influences Sunspot cycles – the position of syzygy between Earth and Venus, when the barycenter of the resonance group most closely approaches the Sun and stops for some time, relative to Jupiter planet, well matches the Sunspot cycle of 11 years, not only for the last 400 years of measured Sunspot cycles, but also in 1000 years of historical record of “severe winters”. We show, how cycles in angular momentum of Earth and Venus planets match with the Sunspot cycle and how the main cycle in angular momentum of the whole Solar system (854-year cycle of Jupiter/Saturn) matches with climatologic data, assumed to show connection with Solar output power and insolation. We show the possible connections between E+V events and Solar global p-Mode frequency changes.
We futher show angular momentum tables and charts for individual planets, as encoded in DE405 and DE406 ephemerides. We show, that inner planets orbit on heliocentric trajectories whereas outer planets orbit on barycentric trajectories.

That looks to me like it might be this paper:

Claims to find an Earth / Venus spin orbit resonance and that it influences solar cycles.

Has a nice long list of many known and some speculated solar cycles.

Shifting the search terms over to “Solar orbital resonance sunspot” you can even find an ‘electric universe’ explanation of what they think is going on:

### In Conclusion

OK, I’m not going to settle anything tonight. Just show you what pops up when using ‘resonance’ instead of spin-orbit coupling. They are not exactly the same things, but related via the angular momentum of the solar system.

And that is the whole point here. To show that the conservation and swapping about of angular momentum in the solar system is not a new idea. It’s fundamental. And it can not be ignored in explaining the behavior of the sun. In my opinion, the “simplified cases” that work well in describing a planet in the context of the sun are not suited to the sun itself. Why? Because it IS so massive, it is NOT a point mass, it is NOT a rigid body, the percentage of change of the radius of rotation is so large, and the external angular momentum that can act on it is so much larger than the percentage of the angular momentum in the sun now. Basically, all the things that make simplifications work for small planets far from the sun make those simplifications wrong in the context of the sun itself.

In the end, just recognizing that spin-orbit coupling is not just for subatomic particles is a big step forward. And it is not just the ‘spin locking’ of a moon to its planets either. Perhaps best put another way. The simple case that says A moon can lock to its planet implies that a nearly chaotic and constantly changing set of forces from ALL the planets acting on the sun will assure it can NOT lock, but will always be a bit ‘off kilter’… but those forces can move mass around inside the sun.

## Dynamics of retrograde (1/n) mean motion resonances: the (1/) , (1/) cases

In this paper, we investigate the dynamics of the exterior retrograde (1/n) resonances within the framework of the planar circular restricted three-body problem (PCRTBP), taking the (1/<-2>) and (1/<-3>) resonances as examples. We prove that there is no asymmetric libration in retrograde (1/n) resonances using analytical, numerical integrations, and semi-analytical methods, by mutual authenticating. For retrograde (1/n) resonances, we calculate the magnitudes of the first- and second-harmonics of the expansion. The analytical results showed that the second-order harmonics in the expansion of the disturbing function can be neglected, which are the leading cause of the existence of the asymmetric libration in prograde (1/n) resonances. Our results also conform to the latest qualitative criterion for the appearance of asymmetric libations proposed by Namouni and Morais (2016). And our analytical results are verified well by numerical integrations. By semi-analytical theory, we generate a series of phase-space portraits in the (e) – (phi ) polar plane for the full range of eccentricity to confirm the absence of the asymmetric librations in retrograde (1/n) resonances. It is the first time the absence of the asymmetric librations in retrograde (1/n) resonances is studied. We also analyze the dynamics of the pericentric and apocentric librations of the retrograde (1/n) resonances. The stable resonant libration zones of the (1/<-2>) , (1/<-3>) , (1/<-4>) and (1/<-5>) resonances are illustrated in the (a) – (e) plane. Our research here reveals the differences between the dynamics of retrograde and prograde (1/n) resonances.

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## A study of the 1/2 retrograde resonance: periodic orbits and resonant capture

We describe the families of periodic orbits in the 2-dimensional 1/2 retrograde resonance at mass ratio (10^<-3>) , analyzing their stability and bifurcations into 3-dimensional periodic orbits. We explain the role played by periodic orbits in adiabatic resonance capture, in particular how the proximity between a stable family and an unstable family with a nearly critical segment, associated with Kozai separatrices, determines the transition between distinct resonant modes observed in numerical simulations. Combining the identification of stable, critical and unstable periodic orbits with analytical modeling, resonance capture simulations and computation of stability maps helps to unveil the complex 3-dimensional structure of resonances.

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## Retrograde periodic orbits in 1/2, 2/3 and 3/4 mean motion resonances with Neptune

We study planar and three-dimensional retrograde periodic orbits, using the model of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and focusing on the dynamics of resonant trans-Neptunian objects (TNOs). The position and the stability character of the periodic orbits can provide important piece of information on the stability and long-term evolution of small TNOs in retrograde motion. Using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 1/2, 2/3 and 3/4 exterior mean motion resonances with Neptune. The bifurcations for planar families of the elliptic model and families of the circular spatial model are determined and the bifurcated families are computed. In our study of the planar elliptic model, we consider the eccentricity of the primaries in the whole interval (0<e'<1) for dynamical completeness. In the spatial circular model, retrograde periodic orbits are obtained mainly from bifurcations of the retrograde planar orbits. Also, we obtain retrograde periodic motion from continuing direct orbits for inclination values larger than (90^circ ) . The linear stability of orbits is of major importance. Generally, stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. TNOs of retrograde motion are not common, but new discoveries cannot be excluded.

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## 2. Hypothesis of a Resonance Between the Earth and Venus's Core

[8] The hypothesis of a spin-orbit resonance between Venus's solid core and the Earth would permit us to understand why the rotation rate of the surface of Venus, as measured by either terrestrial or spaceborne radars, is very close, but not equal to the p = −5 resonance. It requires however a certain number of conditions to be fulfilled. First there should be a solid core with a spin rate different from the mantle's spin rate. Then the core-mantle spin difference should be within the allowable range. Also, the permanent gravitational quadrupole moment created by the solid core should be sufficient to permit the resonance to remain stable, but on the other hand the stresses within the core should not exceed the limit acceptable by the core material. Those questions will be addressed in this section.

### 2.1. Nature of Venus's Core

[9] The similarity in size and density between Venus and Earth suggests that both planets have similar internal structures involving a mantle surrounding a dense iron core. As concerns the Earth, seismological studies have revealed that the core is composed of a liquid outer core surrounding a solid inner core. It is classically believed that the terrestrial magnetic field is produced by dynamo action in the Earth's iron-rich fluid outer core. Fluid motions of the highly conducting liquid in the presence of the magnetic field induce currents which themselves generate the field. The most efficient process for driving the fluid motions in the liquid outer core is chemically driven convection. According to this process, buoyant light material is released as the liquid outer core freezes onto the solid inner core [ Braginsky, 1964 Stevenson et al., 1983 ]. This freezing process at the interface between inner and outer core is therefore believed to be responsible for the generation of convection, which in turn generates the terrestrial magnetic field.

[10] Contrary to the Earth, and despite its similarity in size, Venus does not possess a significant dipole magnetic field. Because of the aforementioned role of the inner-outer core interface in generating the Earth's magnetic field, this lack of a magnetic field at Venus has been interpreted by considering that the Venus core is either entirely liquid [ Stevenson et al., 1983 ] or completely solidified [ Arkani-Hamed and Toksöz, 1984 ]. Konopliv and Yoder's [1996] estimation of the k2 potential Love number from Doppler tracking of Magellan and Pioneer Venus Orbiter, tended to favor the hypothesis of a liquid, rather than solid core for Venus. More recently, however, Stevenson [2003] released those constraints, and concluded that there are two possible reasons why the liquid core of Venus does not convect. The first possibility is that there is no inner core. The other possibility is that the core is currently not cooling, and thus no freezing occurs at the inner core-outer core interface, chemically driven convection being therefore inhibited. According to this approach, such a regime began as Venus transitioned from a mobile surface to a stagnant lid regime, following a resurfacing event about 500 million years ago [ Schubert et al., 1997 ]. Such a difference with the Earth arises because the Earth has plate tectonics, which eliminates heat more efficiently than a stagnant lid form of mantle convection. Also, Greff-Lefftz and Legros [1999] , and Touma and Wisdom [2001] developed models for core differentiated rotation on Venus (or the Earth), and explained how resonance passages in the past may have released energy to produce such resurfacing events of the planet.

[11] On the basis of the preceding discussion, in the remainder of this paper we shall assume that Venus's core consists of an inner and an outer core. While the outer core radius is fairly well known [ Stevenson et al., 1983 ], several assumptions will be tested as concerns the hypothetical inner core radius.

### 2.2. Differential Rotation Between Mantle and Inner Core and Associated Torque

[13] Such a differential rotation rate may be compared to the estimates of differential rotation rate obtained for the Earth by analyzing the travel times of seismic waves traversing the Earth's fluid and solid cores. Most estimates of the terrestrial inner core rotation rate are a few tenths of a degree per year faster than the rotation of the Earth (superrotation). Those estimates are however still rather uncertain, encompassing a large range of values, from zero rotation [ Souriau and Poupinet, 2003 ], to intermediate values of 0.3 to 0.5 degrees/year [ Zhang et al., 2005 ] while estimates of more than 1 degree/year have also been reported [ Song and Richards, 1996 ].

[15] Lacking similar information concerning Venus, we shall use those bounds for the outer core viscosity of Venus. As concerns the outer core radius, Stevenson et al. [1983] reviewed several models of the core with radii ranging between 2890 km and 3110 km. Thus, taking R2 = 3000 km will provide a reasonable order of magnitude. The inner core radius R1 is unknown, and therefore different assumptions will be made here, with R1 = 1500, 2000, and 2500 km, respectively. Table 1 gives the torque TW exerted by the mantle on the core, depending on the assumption made for R1 and η. This may be compared to the amplitude of the torque exerted by the atmosphere on the body, estimated as 1.8 × 10 16 Nm by Dobrovolskis and Ingersoll [1980] under their heating at the ground hypothesis. One can see that, under that hypothesis, the torque from the mantle on the core represents only a very small portion of the atmospheric torque, from 0.04% to 4.2% depending upon the assumption taken for R1 and η in Table 1, most of the torque transmitted by the atmosphere being thus tidally dissipated within the mantle in that case.

R1 = 1500 km R1 = 2000 km R1 = 2500 km
η = 470 Pa s 7.92 × 10 12 Nm 2.34 × 10 13 Nm 7.62 × 10 13 Nm
η = 4700 Pa s 7.92 × 10 13 Nm 2.34 × 10 14 Nm 7.62 × 10 14 Nm

### 2.3. Observations of Gravity and Topography

[16] Both gravity and topography of Venus have been observed extensively by the Pioneer Venus Orbiter and Magellan missions [e.g., Sjogren et al., 1983 Bills and Kobrick, 1985 Bills et al., 1987 McNamee et al., 1993 Nerem et al., 1993 Konopliv et al., 1993 ], and correlations have been made between those measurements. Bills et al. [1987] reported that the correlation coefficient between gravity and topography is well above the 95% confidence level upper bounds on sample statistics from uncorrelated populations for every harmonic degree up to degree 15, except for the lowest degree harmonic (degree 2) for which the correlation coefficient, which is only of the order of 0.3, reveals no statistically significant correlation. This tends to indicate that the internal structure of the planet plays an important role on the n = 2 harmonic of the gravity field. Let us denote A, B, C the inertial moments of Venus, with A < B < C. Konopliv et al. [1993] compared the orientation of the axis of smallest inertia A to the orientations of the principal axes of the ellipsoid that best fits the Venus topography. Estimates of the longitude of axis of smallest inertia A range between −6.5° and −3.0° (hereafter we will take the average −4.7°), depending on the gravity model used. In contrast, among the 3 principal axes of the ellipsoid that best fits the Venus topography, the longest axis (with length 6052.214 km, compared to 6051.877 km and 6051.352 km for the two other axes) points toward a longitude of 281.1°. Orientations of the axes of smallest inertia based on either topography or gravity thus differ by 74.2 degrees. This large difference between the orientations of the gravitational and topographic axes is consistent with the lack of significant correlation between the n = 2 harmonics of topography and gravity. This is an indication that the direction of the axis of smallest inertia A is related to nonuniform density distributions within the planet. It is however not possible to determine whether those nonuniform density distributions are occurring within the mantle, or within the core, or result from nonaxisymmetric boundaries between outer core and mantle, or between inner and outer core. It is likely that the observed quadrupole moment of the planet results from a superposition of contributions from all those possible causes.

### 2.6. Gravitational Coupling Between Core and Mantle

[26] Note that equation (17) was obtained assuming that the liquid outer core has a uniform density (in fact the liquid intermediate medium of Van Hoolst et al.'s [2008] study was an aqueous ocean with uniform density). In contrast, in our case the density of the liquid outer core at its lower boundary is presumably different from ρo due to compressibility of liquid iron, since Dziewonski and Anderson's [1981] terrestrial model gives a density difference as high as 2.5 × 10 3 kg m −3 between the lowermost and uppermost outer core's densities. We will however use equation (17) here for a while because of its simple formulation compared to equation (16), in order to discuss orders of magnitude estimates of the various contributions to the gravity anomaly, taking Δρ as the density difference at the inner core-outer core boundary (Δρ = 0.6 × 10 3 kg m −3 ).

[27] If TCMo is not too strong, the effect of the torque TCM will be an oscillatory behavior of the inner core rotation rate around the resonant rate ωr = 2π/243.1650 days, as will be shown below. Correlatively, the core-mantle differential rate δΩ will oscillate around the average differential rate ΔΩ = 0.31degree/year mentioned in section 2.2.

[30] Thus, it appears that differential rotation between the inner core and mantle will be impossible, unless some very efficient compensation mechanism shields the inner core from the semidiurnal gravity anomalies originating within the mantle. It is proposed here that isostatic compensation, leading to hydrostatic equilibrium below the compensation depth, provides such a mechanism acting to cancel those gravity anomalies in the deep interior. In the simplistic three-layer model described above, the density was assumed constant within the mantle and compensation was expected to be achieved by the shape of the CMB. In reality isostatic compensation is believed to occur principally in the crust (i.e., within the first few tens of km below the planetary surface) and also including parts of the upper mantle, and loads with half width >500 km on Earth are believed to be in approximate isostatic equilibrium [e.g., Keary and Vine, 1996 ]. It should be noted that the assumption of isostatic compensation with an effectiveness of 99.9998%, as discussed above, is a very stringent hypothesis, and the possibility of such high level of compensation is rather speculative. For example, a crust with thickness variations of 30 km having this level of compensation could not sustain an uncompensated load more than 6 cm in amplitude without violating the condition. Such a hypothesis is however needed in the framework of this paper. Of course, this does not exclude the possibility of compensation also occurring at the core-mantle boundary (CMB), and Schubert et al. [2001] propose that broad-scale undulations of the D″ layer (at the base of the mantle) and the CMB probably represent dynamic topography, which would be expected to relax on a short time scale considering the rheology at the high temperatures in D″. Finally, hydrostatic equilibrium is expected to apply within the fluid outer core. Hydrostatic equilibrium below the compensation depth does not however mean that anomalies of the gravitational potential vanish in the deep interior. In this paper, we need to assume that, in the coupled system involving lithosphere, mantle and fluid compressible outer core, the flattening function β(r) will adjust toward a state of minimum energy, which would reduce considerably the gravity anomaly at depth, and correlatively the factor in brackets in equation (16). How high is the residual? This paper will unfortunately not resolve this difficult question, and further realistic modeling of the gravitational potential semidiurnal anomaly below a fully compensated load, involving a multilayer approach including lithosphere, mantle, and radial density gradient in the compressible fluid outer core, would be required to answer that question. Such modeling should compute in a self-consistent manner the radial profile of equatorial flattening β(r) to be used in equation (16).

### 2.7. Consequences on the Balance of the Atmospheric and Body Tidal Torques

[32] Due to these uncertainties of both the atmospheric and body tides, in the following we will use Dobrovolskis and Ingersoll's [1980] heating at the ground estimate (with 100 W m −2 of solar flux absorbed by the ground) To = Tog = 1.8.10 16 Joules as a reference (corresponding to Q = 28), but we will also discuss the consequences if To were reduced by a factor of 100 compared to this estimate (corresponding to Q = 2800).

[35] The equilibrium frequency σe depends only on the relative efficiencies of the atmospheric and body tides, regardless of the positions of the resonance frequencies. Given a value of σe, assumed chosen randomly, the value of parameter λ for the closest resonance (which for the case on hand occurred to be the p = −5 resonance) is necessarily comprised between −0.5 and +0.5, while the probability density for λ is uniformly distributed within the [−0.5, +0.5] interval. Therefore, as soon as λM is larger than 0.5, condition (28) will be fulfilled, without any further condition on the position of equilibrium frequency σe relative to the series of resonances.

[36] Within the heating at the ground hypothesis (To = Tog), this condition λM ≥ 0.5 can be rewritten R1/R2 ≥ 0.43, or equivalently R1 ≥ 1300 km. If R1 < 1300 km, stability of resonance is still possible, but it requires that σe be fortuitously close to a resonance. The probability that the position of σe is compatible with stable resonance is pS = 2λM. For example, from equation (28), probability pS of more than 10% requires R1 ≥ 600 km.

[37] If now we assume that the atmospheric torque is smaller by a factor of 100 than the one obtained with the heating at the ground assumption, then To = Tog/100 and condition (28) yields R1/R2 ≥ 0.094, or equivalently R1 ≥ 280 km. In that case, from equation (26) and assuming the least favorable case ∣λ∣ = 0.5, we obtain that the minimum value of (BA)C for stable resonance should be (BA)C ≥ 2.1 × 10 31 kg m 2 , which is no more than 5.3% of the observed moment difference (BA) of Venus (see section 2.4). Thus in that case even a relatively weak moment difference of the inner core, compared to the moment difference of the whole planet, would be sufficient to ensure stable resonance of the core.

## Astronomy Ch. 8 – The Moon and Mercury

The best way to find the exact distance to the Moon is to:

bounce lasers off the retroreflectors left on the surface by the Apollo landings.

What is the reason that it is so difficult to view Mercury from Earth?

Mercury is always very close to the Sun.

Mercury is very hard to observe from Earth because:

it never gets more than 28 degrees from the Sun’s glare.

From Earth, due to their motions and the fact that the Sun lights only a portion of each surface, both Mercury and the Moon:

appear to go through phases.

Mercury experiences extreme high and low temperatures between night and day because:

it has no atmosphere to moderate temperatures over the globe.

In size, Mercury is intermediate between:

How do the atmospheres of the Moon and Mercury compare?

Neither body has a permanent atmosphere.

If the Earth’s surface temperature were increased to that of Mercury’s day side, then:

we would lose most of our water vapor into space.

Which of the following is NOT a factor in determining whether a body in the solar system retains an atmosphere?

Comparing the densities of the Moon and Mercury, we find:

the Moon’s is similar to Earth’s crust, while Mercury’s is similar to the entire Earth.

One of the effects of Mercury’s very slow spin is

extreme variations in its surface temperature.

almost entirely on Earth side, where the crust was thinner.

more rugged, heavily cratered , and older than the lunar mare.

The youngest features visible with telescopes on the Moon are:

the craters sitting atop the mare.

What is true of the lunar highlands?

They are the oldest part of the lunar surface.

The lunar mare are radioactively dated at:

3.9-3.2 billion years old, forming after most of the bombardment was over.

To measure how Mercury spins, astronomers sent ________ to Mercury and used the Doppler shift to determine how fast it was rotating.

What did radar astronomers find in the polar regions of Mercury?

water ice that never melts in the deep craters

Which statement about the rotations of the Moon and Mercury is FALSE?

Like our Moon, Mercury does not rotate at all, keeping the same side facing the Sun.

How does Mercury’s rotation relate to the Sun?

Its rotation rate is 2/3 as long as its year, due to tidal resonances.

What causes Mercury’s 3:2 spin-orbit resonance?

the planet’s very eccentric orbit the planet’s closeness to the Sun the planet’s high density tidal torques operating on the planet All of the above are factors.

Mercury presents the same side to the Sun

What is true of the Moon’s orbital and rotational periods?

The chief erosive agent now on the Moon is:

the rain of micrometeorites chewing up the regolith.

shows that most interplanetary debris was swept up soon after the formation of the solar system.

The rate of cratering in the lunar highlands shows us that

they range from 4.6 – 4.4 billion years old, on average.

The average rate of erosion on the Moon is far less than on Earth because

the Moon lacks wind, water and an atmosphere.

Which type of feature is the best evidence of lunar volcanism?

rilles associated with lava flows accompanying the mare formation

The spacecraft which reveal the possibility of lunar ice are:

Clementine and Lunar Prospector.

Mercury’s surface most resembles that of which other body?

Which of these features is attributed to the shrinking of Mercury’s core?

Almost all we know about Mercury has come from:

the three flybys of Mariner 10.

Mercury and the Moon appear similar, but we note that:

Mercury has "weird terrain" opposite its huge Caloris basin. Mercury does not always keep the same face toward the Sun, while the Moon does have the Earthside always facing us. the lunar mare are darker than Mercury’s intercrater plains. Mercury has striking lobate scarps due to the shrinking of its core. All of the above are correct.

The scarps on Mercury were probably caused by

the interior cooling and shrinking.

Mercury’s surface most resembles which of these?

Mercury’s Caloris basin is aptly named, since:

it is the hottest region, turning to face the Sun when Mercury is at perihelion.

The Moon’s huge Mare Orientale basin has a twin on Mercury named:

Moonquakes on the Moon were detected by:

the seismographs left there by the Apollo astronauts.

What do moonquakes reveal about the Moon?

Its small, partially molten core has been pulled toward us by tidal forces.

How does Mercury’s magnetic field compare to our own?

It is 1/100th as strong as ours, but does deflect the solar wind to some degree.

What two properties of Mercury imply that it is differentiated?

its large average density and its magnetic field

The presence of a Mercurian magnetic field surprised the planetary scientists on the Mariner 10 team because

the dynamo theory predicted that Mercury was spinning too slowly for one.

Which of these theories seems to best explain the Moon’s origin?

What are the major factors that rule out the co-formation theory for the Moon-Earth system?

Each body has a different density and a different chemical composition.

Which of these would support the capture theory of the Moon’s origin?

the retrograde orbit and large orbital inclination of Neptune’s moon Triton

The cratering of the lunar highlands shows us:

they are older than the smoother maria.

How are the polar regions of Mercury and the Moon similar?

Both seem to have ice pockets in the deepest, darkest crater floors.

Mercury’s evolution was different from the Moon’s because:

dense Mercury had an iron core that shrank, creating the lobate scarps.

Both the Moon and Mercury are geologically inactive and have been that way for most of the history of the solar system. However, about 4 billion years ago, it is thought that

Mercury had more common volcanic activity than the Moon.

Astronomers believe that the Moon did not differentiate to the same degree as Earth because:

the less dense and smaller moon did not have as much radioactivity as the larger Earth in its core.

(SA) Would an observer on Mercury see the Sun rise in the east or the west?

Most of the time, the Sun would appear to move east to west from Mercury’s surface. Near perihelion, however, the Sun goes into retrograde for a few days and moves west to east.

(SA) How is it possible for Mercury and the Moon to have water ice at their poles?

Shadows at the bottom of craters keep sunlight from hitting the ground, so the temperature has always been low enough that ice there has never vaporized and escaped, or that came in with impacting comets and asteroids.

(SA) Why was the discovery of a substantial magnetic field around Mercury a surprise? How was it detected?

In its close passes by Mercury, Mariner 10 found a field that is much stronger than Mercury’s very slow rotation would have led us to expect with the dynamo theory.

(SA) What is the primary source of erosion on the Moon? Why does change there take so long?

A constant fall of meteoroids from space pelts the moon, pulverizing the surface with tiny craters. But really big impacts are rare, and these microscopic changes take a long time to show up as seen from Earth. Our erosive agents like wind, water, and ice can make much more dramatic changes in short periods of time, such as floods, sandstorms, glaciers, etc.

(SA) Relate the formation of Mercury’s scarps to its differentiation.

As Mercury was molten, the dense iron and nickel sank to the core. But smaller Mercury cooled much faster than larger Earth and Venus, so as its core cooled, it also contracted as it solidified. The crust above wrinkled and formed the scarps due to this shrinkage.

(SA) Why has Mercury traditionally been such a hard planet to observe and study?

There are two reasons. First, Mercury is a very small planet, so it never appears very large in telescopes. Second, because it orbits so close to the Sun, it can never be viewed under favorable conditions. It is always seen near the horizon, through a lot of atmosphere which distorts its image. Whenever it is in a favorable position for viewing, so is the Sun!

Allan, R. R.: 1967,Planetary Space Sci. 15, 53 and 1829.

Blitzer, L.: 1965,J. Geophys. Res. 70, 3987.

Blitzer, L.: 1966,J. Geophys. Res. 71, 3557.

Blitzer, L.: 1967,Astron. J. 72, 988 also78, 516.

Colombo, G. and Shapiro, I. I.: 1966,Astrophys. J. 145, 296.

Danby, J. M. A.: 1962,Fundamentals of Celestial Mechanics, The Macmillan Co., New York, p. 310.

Gedeon, G. S.: 1969,Celest. Mech. 1, 167.

Goldreich, P. and Peale, S. J.: 1966a,Nature 209, 1078.

Goldreich, P. and Peale, S. J.: 1966b,Astron. J. 71, 425.

Kaula, W. M.: 1961,Geophys. J. 5, 104.

MacMillan, W. D.: 1930,The Theory of the Potential, McGraw-Hill Book Company, Inc., New York, p. 382.

Plummer, H. C.: 1960,An Introductory Treatise on Dynamical Astronomy, reprinted by Dover, New York, p. 44.

Yionoulis, S. M.: 1965,J. Geophys. Res. 70, 5991 also71, 1289.

## This paper

As the authors honestly admit, it is somehow inaccurate to speak of Hamiltonian formulation when you have dissipation. Their paper deals with the dissipative spin-orbit problem, so their “Hamiltonian” function is not an Hamiltonian strictly speaking, but the ensuing equations have a symplectic structure.

They assume that the dissipation is contained in a function F, which depends on the time t, and discuss the resolution of the problem with respect to the form of F: either a constant dissipation, or a quasi-periodic one, or the sum of a constant and a quasi-periodic one.

Of course, this paper is very technical, and I do not want to go too deep into the details. I would like to mention their treatment of the quasi-periodic case. Quasi-periodic means that the function F, i.e. the dissipation, can be written under a sum of sines and cosines, i.e. oscillations, of different frequencies. This is physically realistic, in the sense that the material constituting the satellite has a different response with respect to the excitation frequency, and the time evolution of the distance planet-satellite and a pretty wide spectrum itself.
In that case, the dissipation function F depends on the time, which is a problem. But it is classically by-passed in assuming the time to be a new variable of the problem, and in adding to the Hamiltonian a dummy conjugate variable. This is a way to transform a non-autonomous (time-dependent) Hamiltonian into an autonomous one, with an additional degree of freedom.
Once this is done, the resolution of the problem is made with a perturbative approach. It is assumed, which is physically realistic, that the amplitudes of the oscillations which constitute the F function are of different orders of magnitudes. This allows to classify them from the most important to the less important ones, with the help of a virtual book-keeping parameter λ. This is a small parameter, and the amplitude of the oscillations will be normalized by λ q , q being an integer power. The largest is q, the smallest is the amplitude of the oscillations. The resolution process is iterative, and each iteration multiplies the accuracy by λ.

It is to be noted that such algorithms are usually written as formal processes, but their convergence is not guaranteed, because of potential resonances between the different involved frequencies. When two frequencies become too close to each other, the process might be destabilized. But usually, this does not happen before a reasonable order, i.e. before a reasonable number of iterations, and this is why such methods can be used. The authors provide numerical tests, which prove the robustness of their algorithm.